Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions

ℒ denotes the Lebesgue measurable subsets of ℝ and ${\displaystyle {ℒ}_0}$ denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ_0${\displaystyle hasaperfect\subset Q\in ℒ\${ℒ}_0}$$ℒ_0!$! which is a subset of or misses M (a similar statement omitting "is a subset of or" characterizes ${\displaystyle {ℒ}_0}$). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the "Marczewski measurable sets" and the σ-ideal ${\displaystyle \left(s^0\right)}$ which we call the "Marczewski null sets". M ∈ (s) if every perfect set P has a perfect subset Q which is a subset of or misses M. M ∈ ${\displaystyle \left(s^0\right)}$ if every perfect set P has a perfect subset Q which misses M. In this paper, it is shown that there is a collection G of ${\displaystyle G_{\delta }}$ sets which can be used to give similar "Marczewski-Burstin-like" characterizations of the collections ${\displaystyle B_w}$ (sets with the Baire property in the wide sense) and FC (first category sets). It is shown that no collection of ${\displaystyle F_{\sigma }}$ sets can be used for this purpose. It is then shown that no collection of Borel sets can be used in a similar way to provide Marczewski-Burstin-like characterizations of ${\displaystyle B_r}$ (sets with the Baire property in the restricted sense) and AFC (always first category sets). The same is true for U (universally measurable sets) and ${\displaystyle U_0}$ (universal null sets). Marczewski-Burstin-like characterizations of the classes of measurable functions are also discussed.